Measuring Wave Phenomena

 

Subject Area

Physics - Engineering

Age or Grade

9-12

Estimated Length

60-70 minutes

Prerequisite knowledge/skills

Initial experience with wave phenomena (qualitative observation with Slinky springs), Newton's 3rd law

Description of New Content

Wave pulses; measurement of wave properties; effects of tension, damping, and amplitude on wave properties such as speed, frequency, wavelength, energy, dissipation.

Reflection at an interface – phase shift on reflection, idea of boundary conditions in Newton's 3rd law.

Interference of wave pulses

Goals

Specifics

1.  Understand what properties are relevant in determining the behavior of a wave, and how to measure those properties, in particular that string tension is the only factor that affects wave speed.

2.  Observe phenomenon of reflection and phase change at an interface – be able to offer a qualitative justification for this behavior in terms of Newton's 3rd law.

3.  Observation of constructive and destructive interference of wave pulses.

4.  Application of universality of wave speeds to a problem of geophysics.

General

From the Massachusetts Frameworks for Science and Technology/Engineering (2006)

Physics: Standards 4.4.1, 4.4.3, 4.4.4

Materials Needed

The PhET demo “Waves on a String” (http://phet.colorado.edu/new/simulations/sims.php?sim=Wave_on_a_String), which requires a java enabled web browser, or can be downloaded to run locally with a java virtual machine (all needed files can be downloaded from the above site). My class had only one working computer, so we did the following exercise as a group discussion/exploration. It could be modified to run as an individual or group lab project if sufficient computers are available.

Procedure

 

 

 

Opener

 

5 min.  Start-up: Ask students to describe in words and pictures how a single point on a string moves as a wave passes along the length of the string.

Development

1.  Start by opening the "Waves on a String" demo, check the option for "Pulses", and click the green "Pulse" button on the screen to set a single wave pulse moving.  Have the students watch it carefully until it disappears (damps out) before sending another pulse.  Ask students if this accurately reflects what they saw with the real springs -- they should generally agree that it is a good simulation.

2.  Ask them to consider why the wave disappears before it reaches the other end -- what must be happening in order to make it disappear? (energy dissipation)

3.  Now ask them to figure out what they could change in the wave's initial settings to make it travel farther.  They will suggest changing the amplitude, the pulse width, the tension, the speed . . .  Show them that we can't control the speed directly, then try altering the amplitude and pulse width and observe the difference in the wave behavior.  There won't be much difference.

4.  Since we couldn't see a significant difference, should we conclude that those variables don't matter?  --- your goal is to have them convince you that some actual measurements would be a good idea.

5.  Show them the use of the horizontal and vertical rulers and the stopwatch, (the orange dotted line at the bottom of the screen can also be moved around to serve as a reference) --  have them propose a plan to use these to determine precisely how far a given wave pulse travels, and also to measure the starting properties of the wave.

6.  Vary individual quantities in turn, showing the effect these have on the pulse travel distance.  They should find that the "Damping" factor has the largest impact -- discuss the meaning of "damping", allowing them to conclude that the damping is the rate of energy dissipation.

7.  Show them that the simulation allows us to turn the damping to Zero.  (Discuss how this is unrealistic, but still useful)

8.  Turn the damping to zero, and watch a wave pulse travel forever.  Have them point out the inversion on reflection, and begin to speculate on why this might happen.    

9.  Change the end of the string to "loose" rather than "fixed", and set a new wave pulse in motion.  The students should immediately see the difference in reflection behavior.  Ask them to speculate on why this is -- specifically on what is different in the given conditions (fixed vs. loose string end) that ensures the reflection behavior.

10.  Leave the reflection question for now, and begin to investigate the speed of the wave pulse, using the rulers and stop-watch to measure a precise speed for different pulses.  Ask the students to figure out which variables will change the speed, since we can't control the speed directly.

11.  They should hit on a strategy of changing one variable, measuring the speed precisely, changing that same variable to a second value, measuring the speed precisely, and comparing.  If two widely different values don't affect the speed, then it is likely that this variable does not affect the speed at all (with more mathematically astute students, discuss what kinds of functional forms might trick us into thinking that a particular variable doesn't matter).

12.  After trying all variables available, they should conclude that the "tension" is the only variable that seems to affect the wave speed.

13. Now explore interference by asking them to predict what will happen when two waves overlap with each other, then demonstrate by sending a pulse down the string with a fixed end, then as the wave hits the fixed end, send a second pulse -- allow them to watch what happens at full speed (with no damping, amplitude relatively high, and tension high as well.  It is better if the pulse width is not too wide, so they can see both waves separately.  Ask them what happens at the precise moment the two waves overlap.

14.  Test their account by slowing the process down -- use the pause button to stop the two waves just before their leading edges make contact, then use the "step" button to allow the two waves to interact one frame at a time -- they will see the two completely cancel for one instant, before appearing to trade places and continue on.  Have them discuss whether the two waves are actually trading places.  They should conclude that before the collision there was a wave on top going to the right and a wave on the bottom going to the left, and that after the collision there is still a wave on top going to the right and a wave on bottom going to the left -- thus it seems that the two waves do not change each other after the pass through, but they seem to affect each other only when they overlap.

15.  Ask them to predict how a single particle on the string moves as the wave passes by it -- most of them will think that that particle first moves forward and up, then back and down to return to its original position.  Use the rulers to prove that an individual point on the string only moves vertically as the wave passes -- thus the direction of energy motion (forward) is different from the direction of particle motion (vertical).

16  Have them use this fact that the particles only move vertically to explain why the two waves seem to cancel each other out only when they overlap (because one wave trys to make a particle move up, but the other tries to make it move down, and the two forces cancel each other out, so the particle remains at rest).

17.  Now switch the end of the string to a loose end, and repeat the wave collision, asking them to predict what will happen when two waves collide on the same side (they are both "up").  Repeat the simulation as above, stepping through one frame at a time to see how the two waves add up when they overlap.

18.  Now that they have seen both kinds of interference, and seen reflection from a loose and a fixed end, ask them to come up with more ideas as to why the loose end does not cause a phase shift, but the fixed end does.

You want them to recognize that a reflection is just like two waves overlapping and interfering with each other.  Show them a loose end reflection in slow motion -- they should notice that for a moment, the amplitude is _twice_ what it was, then show a fixed end reflection -- they should notice that for a moment the amplitude is zero.  Both cases are JUST LIKE a wave collision in the middle of the string.

The answer you want them to come to is that the particle attached to the fixed end cannot move, so the amplitude of the wave MUST BE ZERO there, no matter what else happens.  In order for this to be true, the two overlapping waves (the incident and reflected) must cancel out for a moment, and in order for that to be true, they have to be on opposite sides of the string.  For the loose end, the amplitude of the wave does not have to be zero, so the two waves (the incident and reflected) are able to be of the same phase (both on top).

Another way of looking at this is that the string exerts a force on the end.  When the end is fixed, by Newton's third law, the end is able to exert a balancing force on the string, which pulls it back down.

 

Evaluation

Short homework assignment:  Draw motion of a single particle on the string as a wave passes through it (draw displacement versus time) for the particle. 

Extensions

1.  We know that tension is the only variable that affects the speed of a wave on a string.  We also know that waves can travel through many different kinds of media, including rocks.  Tension in the string is analogous to the mechanical deformation strength of the rock -- show them how this strength can be different in different directions by showing them longitudinal and transverse waves on a long Slinky.  The longitudinal waves travel faster than the transverse, because the spring tension (the restoring force) is stronger in the longitudinal direction.  Tell them that the same is true of rocks, which means that transverse and longitudinal waves will travel at different speeds in a rock.

Challenge question:  Suppose that you begin to feel an earthquake, at noon exactly, as a short and sudden jerk to the north.  Exactly 30 seconds later you feel more motion, this time slower motion side to side from east to west, that lasts for 1 minute.  How far away was the earthquake that caused this motion?

Hints:  They should figure out for themselves that there are two different kinds of waves involved, and that they travel at different speeds, but that the earthquake generated both of them at the same time.  Once they have this, they can ask for the speeds of the two kinds of waves -- 5km/sec and 8km/sec.  They can use this information to calculate the distance to the earthquake source.

Ans: 

1. Since it took 30 seconds for the slower waves to catch up, the faster waves must have been 150 km ahead when they hit (5km/sec*30sec).

2.  It would take 50 seconds for the faster waves to get 150 km ahead, since they travel 3km/sec faster than the slower ones (8-5 = 3, 150 km / 3km/sec = 50 sec.)

3.  In 50 seconds the faster wave would have travelled 400 km (8km/sec * 50 sec = 400 km), so the earthquake must have occurred 400 km away (to the south).

References

http://www.doe.mass.edu/frameworks/scitech/1006.pdf  Massachusetts Science Frameworks

Authors

Mark Betnel, Boston University GK-12 fellow

Erica Wilson, The Engineering School, 12th grade Physics and Engineering teacher

Douglas Dagan, The Engineering School, 12th grade Physics and Engineering teacher